Summary

• Stellar variability can contain information about potential exoplanets.
• But signal is buried amongst other effects: granulation, supergranulation, oscillations, activity.
• Looking to detect QPOs in photometric and/or radial velocity observations.
• Suggest that time-domain analysis with GPs is better than frequency domain with PSDs.
• Simulation study demonstrating this claim.

Methodology

1. Simulate light curve with known PSD parameters.
2. Estimate PSD:
   • Discrete Fourier Transform (FFT)
   • Generalised Lomb-Scargle (GLS)
3. Fit model to PSD (using MCMC after binning).
4. Fit GP to simulated light curve.
5. Compare with known parameters.

PSD Models

Parameters: amplitude \(S_0\), frequency \(\nu_0\), quality (damping) factor \(Q\).

\[P_\textrm{SHO}(\nu) = \frac{2 S_0 \nu_0^4}{(\nu^2 - \nu_0^2)^2 + \frac{\nu^2 \nu_0^2}{Q^2}}\]

Aperiodic

\[P_\textrm{SHO}(\nu)\rvert_{Q = \frac{1}{\sqrt{2}}} = \frac{2 S_0}{1 + \left(\frac{\nu}{\nu_0}\right)^4}\]

c.f., background granulation noise.

Scenarios

1. Single aperiodic component.
2. Single periodic component.
3. Two harmonic (“rotational”) components.
4. Periodic + aperiodic.
5. Two aperiodic components.

Evenly sampled vs. Unevenly sampled.

Data

• Simulated light curves
• 288 days with 100 points per day
• Draw samples from celerite GP
  • periodic + aperiodic terms
  • white noise term
• Subsampled according to timestamps from HARPS-N solar telescope observations between 2015 and 2018.

Windowing Function

Irregularly sampled \(g(t)\) as the convolution of continuous function \(f(t)\) and window function \(w(t)\).

\[g(t) = f(t) \otimes w(t)\]

where \(w(t)\) is approximated by a sum of delta \(\delta()\) functions.

In frequency domain:

\[G(\nu) = F(\nu) \times W(\nu)\]

Windowing

Results: Single aperiodic

Results: Single aperiodic

Results: Single periodic

Results: Two harmonics

Results: Aperiodic + periodic

Results: Two aperiodic

Results

Conclusions

• GP regression in time domain gives better results than PSD modelling in frequency domain.
• Irregular sampling is the nemesis of PSD estimators.
• Revisit recent studies that used PSD estimation for analysis of radial velocities for exoplanet searches.
• Expect similar with PSDs for QPOs in X-ray binaries, AGNs.
• There is a computation cost to GPs

Comments

• Synthetic data generated by GP: not a fair comparison.
• No discussion of uncertainties in the “observed” data.
• Identifiability is an issue for multi-component model.
• Source code is available!
• Do you need to actually recover the parameters accurately?